MATHEMATICS

M. I. VISHIK

ON BOUNDARY-VALUE PROBLEMS FOR QUASILINEAR PARABOLIC SYSTEMS OF EQUATIONS AND ON THE CAUCHY PROBLEM FOR HYPERBOLIC EQUATIONS

(Presented by Academician S. L. Sobolev on 22 V 1961)

In the first part, a mixed boundary-value problem is studied for quasilinear parabolic systems of equations whose spatial part is a strongly elliptic quasilinear operator \((^1)\). In the second part, by means of an analogue of the Galerkin method, the existence of a solution of the Cauchy problem for a linear hyperbolic equation of arbitrary order is proved.

1. Let there be given in the cylindrical domain \(\Omega = G \times (0 < t < T)\) a quasilinear parabolic system

\[ Mu \equiv \frac{\partial u}{\partial t}+ \sum_{|\alpha|,|\gamma|\leq m}(-1)^{|\alpha|}D_\alpha A_\alpha(x,t,D_\gamma u) \equiv \frac{\partial u}{\partial t}+L(u)=h, \tag{1} \]

where \(x=(x_1,\ldots,x_n)\), \(u=(u^1,\ldots,u^N)\), \(h=(h^1,\ldots,h^N)\), \(D_\alpha=\partial^{|\alpha|}/\partial x_{\alpha_1}\ldots \partial x_{\alpha_n}\), \(\alpha=(\alpha_1,\ldots,\alpha_n)\), \(|\alpha|=\alpha_1+\cdots+\alpha_n\), with initial and boundary conditions:

\[ u\big|_{t=0}=\Psi(x),\qquad u\big|_\Gamma=\varphi(x',t),\ldots,\quad D_\omega u\big|_\Gamma=\varphi_\omega(x',t), \tag{2} \]

\(x'\in\Gamma\), \(|\omega|\leq m-1\); \(\Gamma\) is the boundary of \(G\); \(0<t<T\). Suppose that the spatial operator \(L(u)\) in (1), for each \(t\), \(0<t<T\), is strongly elliptic in the sense of \((^1)\), but instead of the definiteness of its variation assumed in \((^1)\), only the semiboundedness of this variation is required, for example in the form

\[ A(w;v,v)\equiv \sum_{|\alpha|,|\beta|\leq m} \bigl(A_{\alpha\beta}(x,t,D_\gamma w)D_\beta v,D_\alpha v\bigr)\geq \]

\[ \geq c^2\left( \sum_{j,\ |\alpha|=m}|D_\alpha w^j|^{\delta_j}|D_\alpha v^j,D_\alpha v^j| \right)-K(v,v), \tag{3} \]

where \(A_{\alpha\beta}=\partial A_\alpha/\partial D_\beta u\); \((\, ,\,)\) is the scalar product with respect to \(x\); \(w(x,t)\), \(v(x,t)\) are arbitrary sufficiently smooth functions, with \(v\in V_0\), i.e. satisfying on \(\Gamma\) the homogeneous boundary conditions (2) (we denote them by \((2_0)\)); the constants \(c^2\) and \(K\) do not depend on \(w\), \(v\), or \(t\). Condition (3) is analogous to condition \((I_1)\) of \((^1)\). Analogously one formulates a condition of type \((I_2)\) of \((^1)\), or another type of semiboundedness condition analogous to item 3 of \((^1)\).

By a solution of problem (1), (2) we understand a function \(u(x,t)\) from the corresponding space \(U\), satisfying conditions (2) and the relation

\[ \left[\frac{\partial u}{\partial t},v\right] +\sum [A_\alpha(x,t,D_\gamma u),D_\alpha v]=[h,v] \tag{4} \]

for every function \(v\in V_0\) (or \(v\in V'_0\), \(V'_0\subset V_0\)); \([\, ,\,]\) is the integral over \(\Omega\). The spaces \(U\) and \(V_0\) are chosen depending on the orders of growth of \(A_\alpha\) with respect to \(D_\gamma u\). For example, \(U\) consists of functions \(u(x,t)\) for which the norms are finite

\[ \left\|\frac{\partial u}{\partial t}\right\|_2 +\sum \|D_\alpha u\|_p<+\infty \qquad (|\alpha|\leq m), \tag{5} \]

where the norm \(\|\ \|\) is taken over \(\Omega\), and \(V_0\) consists of functions \(v(x,t)\) satisfying on \(\Gamma\) the conditions \((2_0)\), with finite norms \(\|D_\alpha v\|_p,\ |\alpha|\leqslant m\).

Let \(f(x,t)\) be a function defined in \(\Omega\) and satisfying the initial and boundary conditions (2). Then, putting \(u=f+z\), where \(u\) is a solution of problem (1), (2) in the sense of (4), we find that \(z\in V_0\) and \(z|_{t=0}=0\); moreover, according to (4),

\[ \left[\frac{\partial z}{\partial t},v\right] +\left[A_\alpha\left(x,t,D_\gamma(f+z)\right)-A_\alpha\left(x,t,D_\gamma f\right),D_\alpha v\right] = \left[h,v\right]-\left[\frac{\partial f}{\partial t},v\right] -\left[A_\alpha\left(x,t,D_\gamma f\right),D_\alpha v\right]. \tag{6} \]

Theorem 1. If \(Lu\) is a semibounded strongly elliptic operator (see (1) and, for example, (3)), then for any conditions (2) and right-hand side \(h(x,t)\) possessing certain smoothness conditions (finiteness of some norms), problem (1), (2) has, and moreover uniquely, a solution.

For brevity we give the idea of the proof only in the special case when \(A_\alpha\) are polynomials in \(D_\gamma u\) of order \(2l+1\) with constant coefficients, and

\[ A(w;v,v)\geqslant \gamma^2 \sum_\beta\left(\sum_\alpha |D_\alpha w|^{2l}D_\beta z,D_\beta z\right) \quad (|\alpha|=m,\ |\beta|=m). \tag{7} \]

The general case is treated analogously, with some complications. In the case under consideration it is required that
\(\dfrac{\partial h}{\partial t},\ \dfrac{\partial}{\partial t}M(f)\in \mathcal L_2(\Omega)\),
\((h-M(f))|_{t=0}=0\),
\(\psi(x)\dfrac{\partial h}{\partial x_i}\),
\(\psi(x)D_\alpha f,\ |\alpha|\leqslant m+2\)*, be summable over \(\Omega\) with a certain power, where \(\psi(x)\) is any function vanishing on \(\Gamma\) together with several derivatives. We note that in other cases the conditions on \(h\) and \(f\) are, as a rule, less restrictive. In \(\Omega\) define the differential operator

\[ B(z)\equiv \Lambda ze^{-\lambda t} -\psi(x)\Delta ze^{-\lambda t} -\frac{\partial}{\partial t}\left[(T-t)\frac{\partial}{\partial t}\left(ze^{-\lambda t}\right)\right], \tag{8} \]

where \(\Lambda\) and \(\lambda\) are sufficiently large numbers. Let \(\{z_k(x)\}\) be a system of functions in \(G\) such that, by means of linear combinations of functions of the form \(B(z_k(x)d_k(t))\), where \(d_k(t)\) are arbitrary smooth functions of \(t\), \(d_k(0)=0\), one can approximate in the norm
\(\|v\|=\sum\|D_\alpha v\|_{2l+2},\ |\alpha|\leqslant m\),
any function \(v\in V_0\subset W^{m}_{2l+2}\). The existence of such a system \(\{z_k(x)\}\) is easily established by the method of separation of variables with respect to \(x\) and \(t\) for the operator \(B\) and Lemma 2 of (1). The approximate solution of problem (1), (2), or, equivalently, (6), is sought in the form
\(\widetilde z_n(x,t)=\sum C_{in}(t)z_i(x),\ i\leqslant n\), and the coefficients \(C_{in}(t)\) are determined from the following moment equations:

\[ \left(B^*\frac{\partial \widetilde z_n}{\partial t},z_k(x)\right) +\left(B^*\left(L(f+\widetilde z_n)-L(f)\right),z_k(x)\right) = \left(B^*(h-M(f)),z_k(x)\right) \tag{9} \]

under the boundary conditions

\[ C_{in}(0)=C'_{in}(0)=0\quad (i=1,\ldots,n);\qquad C_{in}|_{t=T}\ \text{is bounded.} \tag{10} \]

(9) is a system of ordinary equations of third order in \(t\), degenerating at \(t=T\).

* These conditions on \(f(x,t)\), obviously, impose certain restrictions on the smoothness of the boundary and initial data in (2) and on their compatibility on \(\Gamma\) at \(t=0\).

Lemma 1. Problem (9), (10) has at least one solution \(C_{in}(t)\). For the corresponding approximations \(\tilde z_n(x,t)\) the norms are uniformly bounded (in the case (7)):

\[ \|D_\alpha \tilde z_n\|_{2l+2} + \left\|\frac{\partial \tilde z_n}{\partial t}\right\|_2 \le K, \qquad \left\|\frac{\partial}{\partial x_i}(D_\alpha \tilde z_n)^{\,l+1}\right\|_2' + \left\|\frac{\partial}{\partial t}(D_\alpha \tilde z_n)^{\,l+1}\right\|_2' \le K, \quad |\alpha|\le m, \tag{11} \]

where \(\|\ \|\) is the norm taken over \(\Omega\), and \(\|\ \|'\) is over \(\Omega'=G'\times(0<t<T)\), \(\overline{G'}\subset G\).

The first part of the lemma is proved by reducing the problem to a system of integral equations with respect to \(\partial C_{in}/\partial t\); moreover, the corresponding operator has the form \(E+V\), where \(V\) is a completely continuous operator. It is shown that on a sphere of sufficiently large radius in the corresponding Hilbert space the index of the covering of zero under the mapping \(E+V\) is equal to one. To prove the estimates (11), we multiply both parts of (9) by \(C_{in}(t)\), sum over \(i=1,\ldots,n\), and integrate with respect to \(t\). Next we use estimate (5) (respectively (6)) from \((^1)\) and the estimate (7) given here. From the inequality thus obtained we derive the estimates (11). It follows from (11) that one can choose a subsequence from \(\{\tilde z_n\}\) (we denote it also by \(\{\tilde z_n\}\)) for which, weakly,
\[ \partial \tilde z_n/\partial t \to \partial z/\partial t,\qquad (D_\alpha \tilde z_n)^r \to (D_\alpha z)^r,\quad r\le 2l+2, \]
and moreover \(D_\alpha \tilde z_n\) converges almost everywhere to \(D_\alpha z\) (see \((^1,^2)\)), \(|\alpha|\le m\).

We multiply both parts of (9) by arbitrary functions \(d_k(t)\), \(d_k(0)=0\), integrate with respect to \(t\) from \(0\) to \(T\), transfer \(B^*\) to the second factor, and pass to the limit as \(n\to\infty\). We obtain relation (6), in which \(B(z_k(x)d_k(t))\) is substituted for \(v\). Since by linear combinations of such functions one can approximate any function \(v\in V_0\), the existence of the desired function \(z(x,t)\) is proved, and with it also of the function
\[ u(x,t)=f(x,t)+z(x,t), \]
which solves problem (1), (2).

The uniqueness of the solution obtained is established analogously to \((^2)\). We note that, by virtue of the last estimates (11), the constructed function \(z(x,t)\), and together with it the function \(u(x,t)\), admits derivatives of order \((m+1)\) inside \(\Omega\).

2. Cauchy problem for hyperbolic equations. Let

\[ a(u)=h \tag{12} \]

be a normal hyperbolic equation in the sense of I. G. Petrovskii (see \((^4{}^{-}{}^6)\)) of order \(m+1\), with coefficient of \(\partial^{m+1}u/\partial t^{m+1}\) equal to one. For brevity we prescribe homogeneous initial conditions:

\[ u\big|_{t=0}=0,\ldots,\left.\partial^m u/\partial t^m\right|_{t=0}=0. \tag{13} \]

According to a well-known remark of I. G. Petrovskii \((^4)\), it is enough to prove the existence of a solution of problem (12), (13) under periodic boundary conditions:

\[ D_\beta u\big|_{x=0}=D_\beta u\big|_{x=2\pi},\qquad |\beta|\le m. \tag{14} \]

Let \(b(u)\) be an operator separating \(a(u)\) in the sense of Leray \((^5)\) \(\bigl(b(u)=\partial a(u)/\partial D_tu\bigr)\). The existence of a solution of problem (12), (13), (14), for example for \(0<t<T\), can be established with the aid of an analogue of Galerkin’s method. To this end, we seek the approximate solution \(u_r(x,t)\) of the problem in the form
\[ u_r(x,t)=\sum C_{\alpha r}(t)z_\alpha(x), \]
where
\[ \alpha=(\alpha_1,\ldots,\alpha_n),\qquad |\alpha|\le r,\qquad r=1,2,\ldots, \]
\[ z_\alpha(x)=\exp i(\alpha_1x_1+\cdots+\alpha_nx_n). \]
The coefficients \(C_{\alpha r}(t)\) are determined so that

\[ [\varphi(t)a(u_r),\, b(z_\gamma(t)d_\gamma(t))] = [\varphi(t)h(x,t),\, b(z_\gamma(x)d_\gamma(t))], \qquad |\gamma|\le r, \tag{15} \]

where
\[ \varphi(t)=e^{-\lambda t}-e^{-\lambda T}\quad (0<t<T); \]
\(\lambda\) is a sufficiently large number; \(d_\gamma(t)\) are arbitrary smooth functions of \(t\) satisfying homogeneous conditions at \(t=0\) ...

(13). Conditions (15) lead to a system of order \((2m+1)\) with respect to \(C_{\alpha r}(t)\):

\[ \bigl(b^*(\varphi a(u_r)), z_\gamma(x)\bigr) = \bigl(b^*(\varphi h), z_\gamma(x)\bigr) \tag{16} \]

and to the boundary conditions

\[ C_{\alpha r}^{(s)}(0)=0 \qquad (s=0,1,\ldots,m), \]

\[ \left. \left( \frac{\partial}{\partial t}\bigl(\varphi(t)(a(u_r)-h)\bigr), z_\gamma \right) \right|_{t=T} =0,\ldots, \]

\[ \left. \left( \frac{\partial^{m-1}}{\partial t^{m-1}}\bigl(\varphi(t)(a(u_r)-h)\bigr), z_\gamma \right) \right|_{t=T} +\cdots=0 \tag{17} \]

and to conditions of boundedness of the derivatives for \(t=T\) (the system degenerates at \(t=T\)). The boundary conditions (17) for \(t=T\) arise as conditions for the vanishing of the coefficients of \(\partial^s d_\gamma(t)/\partial t^s\big|_{t=T}\) when the operator \(b\) is transferred to the first factor in (15).

Lemma 2. For sufficiently large \(\lambda\), the boundary-value problem (16), (17) has a solution, and moreover a unique one.

Lemma 3. If the coefficients of the operator \(a(u)\) are sufficiently smooth, then \(u_r\), together with the derivatives \(D_t^lD_\alpha u_r\), \(l\le m\), \(l+|\alpha|\le m+p\), have norms uniformly bounded with respect to \(r\): \(\|D_t^lD_\alpha u_r\|\le k\).

From the sequence \(u_r\) we select a subsequence (denote it also by \(u_r(x,t)\)) which converges weakly to \(u(x,t)\), with \(D_t^lD_\alpha u_r \to D_t^lD_\alpha u\). In (15), in the term with the factor \(\partial^{m+1}u_r/\partial t^{m+1}\), we transfer one derivative with respect to \(t\) to the factor \(b(z_\gamma d_\gamma)\), and, for fixed \(\gamma\), pass to the limit as \(n\to\infty\). From the resulting relation we infer that \(u(x,t)\) also has a derivative of the form \(\partial^{m+1}u/\partial t^{m+1}\) \((\in L_2(\Omega))\) and satisfies the equation \(a(u)=h\). (Here one uses the lemma that linear combinations of functions of the form \(b(z_\gamma d_\gamma)\) are dense in the corresponding space.) From the equation \(a(u)=h\), which is satisfied by the constructed function \(u(x,t)\), and from Lemma 3, we infer that \(u(x,t)\) also has derivatives of the form \(D_t^{l_1}D_\alpha u\) for all \(|\alpha|+l_1\le m+p\).

Theorem 2. The sequence of approximations \(u_r(x,t)\), obtained by solving the system of ordinary equations (16) (of order \(2m+1\)) under the boundary conditions (17), converges strongly (together with the derivatives of order \((m-1)\)) to the solution \(u(x,t)\) of problem (12), (13), (14). If the coefficients \(a(u)\) and \(h(x,t)\) are sufficiently smooth, the function \(u(x,t)\) admits, respectively, derivatives of higher orders.

The uniqueness of \(u(x,t)\) follows from the energy inequality. By slightly modifying the construction of the approximations \(u_r(x,t)\), one can improve their convergence to \(u(x,t)\).

Received
16 V 1961

CITED LITERATURE

1. M. I. Vishik, DAN, 138, No. 3 (1961).
2. M. I. Vishik, DAN, 137, No. 3 (1961).
3. S. L. Sobolev, Some Applications of Functional Analysis to Mathematical Physics, L., 1950.
4. I. G. Petrovsky, Matem. sbornik, 2 (44), 5 (1937).
5. J. Leray, Lectures on Hyperbolic Equations with Variable Coefficients, Princeton, 1952; L. Gårding, The Cauchy Problem for Hyperbolic Equations, IL, 1961.